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Unformatted text preview: 944181T .. mnc. Area under the Stumhud Normal] Curve
MATHS 8‘ HONG KONG EXAMINATIONS AUTHORITY   —
STAT HONG KONG ADVANCED LEVEL'EXAMINATION 1994 3"" "1' “’2 '0’ ""1 "’5 '0“ '°7 “’1 ""’ .
.0000 .0040 .0080 .01 211 .1) 1 (311 .11 I 99 .0239 .0279 .0319 0035‘) .0398 .0438 .0478 .0517 .0557 0596 .0636 .0675 .0714 .0753
.0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
.1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
.1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 MATHEMATICS AND STATISTICS ASLEVEL .1915 I ".1950 .1985 .2019 .2054 .2088 .2123 _.2157 .2190 .2224
.2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
.2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 t ,2823 .2852
.2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 1.3106 .3133
r .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 9.00 ram12.00 noon (3 hours) ~
This baper must be answered in English .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621
.3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 13810 .3830 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015
.4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 ..4306 .4319 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .44 111 .4429 .4441
.4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
.4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
.4713 .4719 ‘ .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 1. This paper consists of Section A and Section B. 2. Answer ALL questions in Section A, using the AL(C1) answer boOk. .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817
.4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857
.4861 .4864 .4868 .4871 .4875 .4878 .4881 T4884 .4887 .4890
.4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 _ .4913 .4916
.4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932, .4934 .4936 3. Answer any FOUR questions in Section B, using the_AL(C2) answer book. 4. Un1ess otherwise speciﬁed, numerical answers should either be exact or given to 4 dx'mﬂi p1aces. .4933 .4940 I .4941 .4943 .4945 .4946 .4943 .4949 .4951 .4952
.4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 = .4963 .4964
.4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 . .4973 .4974
.4974 .4975 . .4976 .4977 . .4977 .4973 .4979 .4979 .4930 . .4981
.4981 .493; .4982 .4983 .4984 .4934 .4985 .4935 .4986 .4986 .4987 .4987 '.4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990
.4990 .4991 .4991 .4991 .4992 ‘ .4992 .4992 .4992 .4993 .4993
.4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 5 .4995 .4995
.4995 .4995 .4995 .4996 .4996 V .4996 .4996 .4996 .4996 .4997
.4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 Note : An entry in the table is the propohion of the area under the entire curve which is between
‘2 = 0 and a positive value 01.7. Areas for negative values of: are obtained by symmetry. .1‘ 1 Atz‘) z 1 ‘'
A(z) = f ———e 1d,:
0 .727: 94ASLMATHS a. STAT—1 1
94»ASLMATHS & STAT18 SECTION A (40 marks)
Answer ALL questions in this section.
Write your answers in the AL(C1) answer book. 1.‘ ' (3) Write down the sample space of the sex patterns of the children of a
2child family in the order of their ages. (You may use B to denote a boy and G to denote a girl.) (b) Assume_that having a boy or having a girl is equally likely. It is
known that a family has two children and they are not both girls. (i) Write down the sample space of the sex patterns of the
children in the order of their ages. ' (ii) What is the probability that the family has two sons?
(4 marks) 2. The population size x of an endangered species of animals is modelled by
the equation ‘ 2
ﬂ—Zi‘l—3x=0, d:2 dt where t denotes the time. It is known that x = 100 e“ where k is a negative constant. Determine
the value of k . (5 marks) 94ASL—MATHS & STAT2 20 3. Jack climbs along a cubical framework from a corner A to meet Jill at the
opposite comer B . The framework, shown in Figure l, is formed by
joining bars of equal length. Jack chooses randomly a path of the shortest
length to meet Jill. An example of such a path, which can be denoted by Right  Up  Forward  Up  Right  Forward , is also shown in Figure 1. Forward Rig/1!
Figure]
(3) Find the number of shortest paths from A to B . (b) If there is a trap at the centre C of the framework which catches
anyone passing through it, (i) (ﬁnd the number of shortest paths from A to C , (ii) hence ﬁnd the probability that Jack will be caught by the trap on his way to B . =
(5 marks) t 94ASLMATHS & STA'i3 21 4, Figure 2 shows the cumulative frequency polygon of .ghts (in kg) for a
’ group of 100 students. Cumulative Frequency Figure 2 Cumulative frequency polygon of weights
for a group of 100 students (3) Use the graph paper on Page 4 to draw a histogram of the weights. (1)) Determine the interquartile range of the weights from the
cumulative frequency polygon. (0) Determine the mean weight from the histogram.
(6 marks) S4—ASLMATHS 8: STAT4 22 Page Total 4.(Cont‘d) Fill in the details in the ﬁrst three boxes above and tie this sheet
INSIDE your amwer book. Weight (kg) 94—ASLMATHS 5i STAT5 23 This is a blank page The rate of spread of an epidemic can be modelled by the equation
—  3; 12+: , where x is the number of people infected by the epidemic and t is the
number of days which have elapsed since the outbreak of the epidemic. If x =10 when (=0 , express x in terms of I.
(6 marks) (a) Use the exponential series to ﬁnd a polynomial of degree 6 which I approximates e 2 for x close to 0.
1 1:
Hence estimate the integral f e 2dr.
' 0 (b) It is known that the area under the standard normal curve between 1
a 1 m! z=0 and z=a is me 1:12. Use the resnlt of(a) and
0 2n
the normal distribution table to estimate, to 3 decimal places, the value of r: .
(7 marks) In asking some sensitive questions such as "Are you homosexual?", a
randomized response technique can be applied: The interviewee will be
asked to draw a card at random from a box with one red card and two
black cards and then consider the statement ‘I am homosexual‘ if the card
is red and the statement ‘I am not homosexual‘ otherwise. He will give the
response either ‘True' or ‘False'. The colour of the card drawn is only
known to the interviewee so that nobody knows which statement he has responded to. Suppose in a survey, 790 out of 1200 interviewees give
the response ‘True'. (a) Estimate the percentage of persons who are homosexual.
(b) For an interviewee who answered ‘True‘, what is the probability that he is really homosexual?
(7 marks) 94ASLMATHS & STAT6 24 ( 94ASL~MATHS a. STA‘tI 25 SECTION B (60 marks) Answer any FOUR questions from this section. Each question carries 15 marks.
Write your answers in the AL(C2) answer book. 8. Consider the curve
C y = x + 1 (x at 2) .
x ~ 2 
. d y
a Find — .
( ) dx
(2 marks)
(1)) Find the equations of the horizontal and vertical asymptotes to the (C) ((1) curve C .
(3 marks) Sketch the curve C and indicate the asymptotes and intercepts.
(5 marks) Find the area of the region bounded by the curve C , the negative
x—axis and the negative yaxis. I
(5 marks) 94ASL—MATHS & STAT8 26 9. A textile factul‘y plans to install a weaving machine on lst January 1995 to increase its production of cloth. The monthly output 1: (in km) of the
machine, after t months, can be modelled by the function x = 100e'°'°”—' 65 e‘°'°2'  35.
(a) (i) In which month and year will the machine cease producing
any more cloth ? i
(ii) Estimate the total amount of cloth, to the nearest km,
produced during the lifespan of the machine. (5 marks) (b) Suppose the cost of producing 1 km of cloth is US$300; the monthly
maintenance fee of the machine is US$300 and the selling price of
1 km of cloth is US$800. In which month and year will the greatest
monthly proﬁt be obtained? Find also the proﬁt, to the nearest
USS, in that month. (6 marks) (0) The machine is regarded as ‘inefﬁcient’ when the monthly proﬁt
falls below US$500 and it should then be discarded. Find the
month and year when the machine should be discarded. Explain
your answer brieﬂy. (4 marks) 94ASLMATHS & srt ) 27 lO.  A chemical plant discharges pollutant to a lake at an unknown rate of r(t) units per month, where l is the number of months that the plant has been
in operation. Suppose r(0) = 0 . The government measured r‘(t) once every two months and reported the
following ﬁgures: ‘ (a) Use the trapezoidal rule to estimate the total amount of pollutant
which entered the lake in the ﬁrst 8 months of the plant’s operation.
‘ (2 marks) (b) An environmental scientist suggests that r(t) = m” ,
where a and b are constants. (i) Use the graph paper on Page 10 to estimate graphically the
values of a and b correct to 1 decimal place. (ii) Based on this scientist‘s model, estimate the total amount of pollutant, correct to 1 decimal place, which entered the lake in the ﬁrst 8 months of the plant’s operation.
‘ ' (8 marks) (c) It is known that no life can survive when 1000 units of pollutant
have entered the lake. Adopting the scientist’s model in (b), how
long does it take for the pollutant from the plant to destroy all life in the lake? Give your answer correct to the nearest month.
(5 marks) 941ASL—MATHS & STAT10 28 Page Total 10.(Cont‘d) If you attempt Question 10, ﬁll in the details in the ﬁrst three
boxes above and tie this sheet INSIDE your answer book. 94—ASL—MATH5 & STAT11 29 11. A day is regarded as humid if the relative humidity is over 80% and is
regarded as dry otherwise. In city K, the probability of having a humid day is 0.7 . (3) Assume that whether a day is dry or humid is independent from day
to day. (i) Find the probability of having exactly three dry days in a
week (7 days). (ii) What is the mean number of dry days before the next humid
day? Give your answer correct to 3 decimal places. (iii) Today is dry. What the probability of having two or more
humid days before the next dry day?
(8 marks) (b) After some research, it is known that the relative humidity in city K
depends solely on that of the previous day. Given a dry day, the
probability that the following day is dry is 0.9 and given a humid This is a blank page day, the probability that the followmg day is humid is 0.8 .
(i). If it is dry on March 19, what is the probability that it will
be humid on March 20 and dry on March 21? (ii) If it is dry on March 19, what is the probability that it will
be dry on March 21? (iii) Suppose it is dry on both March 19 and March 21.‘ What is
the probability that it is humid on March 20?
(7 marks) 94ASL—MATHS & STAT12 ‘
' 30 94ASL—MATHS & STAT13 3 I 12. Table 1 shows the number of bankdrafts sold per working day in a certain
branch of a bank. Table I Number of bankdrafts sold Number of bankdrafts sold __.—
mwamooqaé 'OOQQUIJiLAN—e (a) Itis suggested that the number of bankdrafts sold follows a Poisson distribution with mean 4.2 . Fill in the expected frequencies, under
this distribution, in column 3 of Table 2 (Page l4). (3 marks) (b) Another suggestion is that the number sold can be approximated by a normal distribution with mean u. and variance :12 . Using class
intervals 0.5 — 1.5 , 1.5 — 2.5 , , 7.5 — 8.5 , some expected frequencies are calculated and shown in the fourth column
of Table 2. Given that the expected frequency for the class oo — 1.5 is 3.340 , determine p and 02 . Fill in the other
expected frequencies. (6 marks) (0) Compare the observed and the two expected frequency distributions
by drawing histograms on the graph paper on Page 15. Hence determine which of the two distributions is more suitable for ﬁtting
the observed data. (4 marks) (d) Find the probability that the branch will sell 4 or more bankdrafts
on a working day under .(i) the model of Poisson distribution in (3); (ii) the model of normal distribution in (b).
(2 marks) 94ASLMATHS s. STAT14 32 Page Total 12.(Cont‘d) If you attempt Question 12, ﬁll in the details in the first three
boxes above and tie this sheet INSIDE your answer book. Table 2 Observed and expected frequencies of bankdrafts sold Expected Frequency Normal Number of
Bankdrafts Sold Frequency 3.149 2.200 m 9.258
10  4.595 AWN LII t
94ASLMATHS & $.AT15 33 13. Batches of screws are produced by a manufacturer under two different sets
of conditions, favourable and unfavourable. If screws are produced under favourable conditions, the diameters of the screws will follow a normal*
12.(Cont’d) > “Re Tm“ distribution with mean 10 mm ' and standard deviation 0.4 nim .. lf screws are produced under unfavourable conditions, the diameters of the
screws will follow a normal distribution with mean 12.3 mm and standard
deviation 0.6 mm . A batch of screws is examined by measuring the
diameter X mm of a screw randomly selected from the batch. (a) The batch is classiﬁed as acceptable by the manufacturer if X < c1 and as unacceptable if otherwise. The value C1 satisﬁes P(X < ct) = 0.95 under favourable conditions. Determine the value of t:1 . I (3 marks) (b) The buyer u5es a different criterion instead. He classiﬁes the batch
as acceptable if X< 02 and as unacceptable if otherwise. The value (:2 satisﬁes P(X<cz) = 0.01 under unfavourable conditions. Determine the value of c2 . (3 marks) Frequency (c) For a batch of screws produced under favourable conditions and
based on the same measurement of a screw, ﬁnd the probability that
the batch will be classiﬁed as unacceptable by the manufacturer, but
acceptable by the buyer. (4 marks) (d) After some negotiation, the manufacturer and the buyer agree to use
a common cutoff point c3 such that P(X< 03) under favourable conditions is equal to P(X 2 63) under unfavourable conditions. Determine the value of £3 . ' (3 marks) (e) The manufacturer and the buyer later agree that a batch will be
rejected in the future if x> 10.8 (too thick) or x< 9.4 (too thin).
If the population mean u. nun of the diameters of the screws
produced can be modiﬁed by adjusting the machine, ﬁnd u so that the probability of rejection, P(X< 9.4 or X> 10,8) , is minimized.
(2 marks) Class mark END OF PAPER 94'ASLMATHS & STAT16 . 34 SLASLVMATHS & STAT_17 35 ...
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